Spectral Calculations for Schroedinger Operators with Quasiperiodic Potentials
Time and Place
Thursday, February 15, 2018
Tea will be served at 3:30 PM in PHSC 424.
Schroedinger operators with quasiperiodic potentials arise as mathematical models of quasicrystals. These operators are fascinating: in many cases the spectrum is a Cantor set, which has important implications for dynamics. This Cantor structure poses a significant challenge for numerical approximations of the spectrum. Periodic approximations play a critical role; via Floquet theory, their spectra comprise the union of real intervals whose size generally diminishes as the period increases. Good approximations to quasiperiodic models require long periods, again challenging attempts at good numerical work. Models in higher dimension remain considerably more challenging and only now are coming within reach. Even simple models composed from one-dimensional models (giving a spectrum that is the sum of Cantor sets) are difficult to understand. In this talk I will address a variety of questions relating to spectral calculations for discrete Schroedinger operators. We will show how to efficiently approximate Cantor spectrum for one-dimensional models (and the limits of these techniques), and explore some intriguing questions about the band structure of higher-dimensional models. This talk describes joint work with David Damanik, Jake Fillman, and Anton Gorodetski.
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